Direct and inverse problems for a damped string

Abstract

In this paper small transverse vibrations of a string of inhomogeneous stiffness in a damping medium with the left end fixed and the right end equipped with a concentrated mass are considered. By means of the Liouville transformation the corresponding differential equation is reduced to a Sturm–Liouville problem with parameter-dependent boundary conditions and parameter-dependent potential. This problem is considered as a spectral problem for the corresponding quadratic operator pencil. The inverse problem, i.e. the determination of the potential and the boundary conditions by the given spectrum and length of the string, is solved for weakly damped strings (having no purely imaginary eigenvalues). Uniqueness of the solution in an appropriate class is proved.